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In this paper, we address the trajectory planning problem in uncertain
nonconvex static and dynamic environments that contain obstacles with
probabilistic location, size, and geometry. To address this problem, we provide
a risk bounded trajectory planning method that looks for continuous-time
trajectories with guaranteed bounded risk over the planning time horizon. Risk
is defined as the probability of collision with uncertain obstacles. Existing
approaches to address risk bounded trajectory planning problems either are
limited to Gaussian uncertainties and convex obstacles or rely on
sampling-based methods that need uncertainty samples and time discretization.
To address the risk bounded trajectory planning problem, we leverage the notion
of risk contours to transform the risk bounded planning problem into a
deterministic optimization problem. Risk contours are the set of all points in
the uncertain environment with guaranteed bounded risk. The obtained
deterministic optimization is, in general, nonlinear and nonconvex time-varying
optimization. We provide convex methods based on sum-of-squares optimization to
efficiently solve the obtained nonconvex time-varying optimization problem and
obtain the continuous-time risk bounded trajectories without time
discretization. The provided approach deals with arbitrary (and known)
probabilistic uncertainties, nonconvex and nonlinear, static and dynamic
obstacles, and is suitable for online trajectory planning problems. In
addition, we provide convex methods based on sum-of-squares optimization to
build the max-sized tube with respect to its parameterization along the
trajectory so that any state inside the tube is guaranteed to have bounded
risk.